Infinite Series
Concepts for Infinite Series  JEE and Board Exams
Concepts:

Convergence and Divergence:
 An infinite series converges if its partial sums approach a finite limit, while it diverges if the partial sums tend to infinity.

Ratio Test:
 If ( \lim\limits_{n\to\infty} \left \dfrac{a_{n+1}}{a_n} \right = L ) exists, then the series ( \sum a_n ) converges if ( L < 1 ), and diverges if ( L >1 ) or ( L ) does not exists.

Root Test:
 If ( \lim\limits_{n\to\infty} \sqrt[n]{  a_n } = L ), then the series ( \sum a_n ) converges if ( L <1 ), and diverges if ( L >1 ).

Comparison Tests:
 Limit Comparison Test:
 If ( \lim\limits_{n\to\infty} \dfrac{a_n}{b_n} = L ), where ( L ) is a finite nonzero constant, then ( \sum a_n ) and ( \sum b_n ) either both converge or both diverge.
 Ratio Comparison Test:
 If ( \lim\limits_{n\to\infty} \dfrac{a_n}{b_n} = L ), where ( L ) is a finite nonzero constant, then ( \sum a_n ) and ( \sum b_n ) either both converge or both diverge.
 Limit Comparison Test:

Alternating Series Test:
 If ( a_{n+1} \le a_n ), and ( \lim\limits_{n\to\infty} a_n = 0 ), then ( \sum (1)^n a_n ) converges.

Integral Test:
 If ( f(x) ) is continuous, positive, and decreasing on the interval ( [k, \infty) ), where ( k ) is an integer, then the series ( \sum\limits_{n=k}^\infty f(x) ) and the integral ( \int\limits_k^\infty f(x) dx) either both converge or both diverge.

pSeries:
 The series ( \sum\limits_{n=1}^\infty \dfrac{1}{n^p} ) converges if ( p>1 ) and diverges if ( p\le1 ).

Telescoping Series:
 A series of the form ( \sum\limits_{n=1}^\infty (a_{n+1}  a_n) ) is called a telescoping series. It can be evaluated by simplifying the expression inside the parentheses, resulting in a convergent series.

Cauchy’s nth Root Test:
 If ( \lim\limits_{n\to\infty} a_n^{1/n} = L ), then the series ( \sum a_n ) converges if ( L<1 ) and diverges if ( L>1 ) or (L) does not exists.